aa r X i v : . [ m a t h . R T ] F e b A CATEGORIFICATION OF THE CARTAN-EILENBERGFORMULA
JUN MAILLARD
Abstract.
We prove a categorification of the stable elements formula of Car-tan and Eilenberg. Our formula expresses the derived category and the stablemodule category of a group as a bilimit of the corresponding categories for the p -subgroups. A classical theorem by Cartan and Eilenberg [CE56, Theorem 10.1] presentsthe mod p cohomology of a finite group G as a subalgebra of the cohomology ofany p -Sylow subgroup S of G . The formalism of fusion systems provides a compactformula to express this result in terms of the fusion category F S ( G ) (definition 2.1): H ∗ ( G ; F p ) ≃ lim P ∈F S ( G ) op H ∗ ( P ; F p ) This formula actually holds [Par17] for any cohomological Mackey functor M : M ( G ) ≃ lim P ∈F S ( G ) op M ( P ) We prove a categorified Cartan-Eilenberg formula (theorem 3.9) for any p -monadicMackey -functor M (definitions 3.2 and 3.7): M ( G ) ∼ = bilim P ∈ ˆ T S ( G ) op M ( P ) For instance, the left-hand side can stand for the categories of group representa-tions M ( G ) = mod ( k G ) , the stable categories of group representations M ( G ) = stmod ( k G ) or the derived categories of the group algebras M ( G ) = D b ( k G ) . (Thereare other examples, see remark 3.8.) We can let G range over all finite groupoids andthese mappings then describe the values on objects of -functors M : gpd f → Cat from the -category of finite groupoids, with faithful functors as -morphisms, tothe -category of small categories.This categorification requires us to replace the fusion category F S ( G ) by anextended transporter category ˆ T S ( G ) (definition 2.4) and the classical limit by apseudo bilimit (definition 1.5) taken in the -category of categories.Through a 2-finality argument, using the criterion of [Mai21], the bilimit intheorem 3.9 can be reinterpreted as a descent-shaped bilimit. Theorem 3.9 thusstates that the functor M is a -sheaf, which allows us to recover the main theoremof [Bal15]. Details on this point of view will appear in my thesis [Mai]. Mathematics Subject Classification.
Key words and phrases.
Categorification, fusion of finite groups, bilimits.Author supported by Project ANR ChroK (ANR-16-CE40-0003) and Labex CEMPI (ANR-11-LABX-0007-01).
The article is organized as follows. In section 1, we recall classical definitionsconcerning bicategories and relevant notations. In section 2, we give a quick sum-mary of the classical categorical formalism of fusion in groups and introduce ourgeneralization. In section 3, we state and prove the main theorem. In section 4, weexplain how to retrieve the classical Cartan-Eilenberg formula, and similar results,from our categorified formula.
Acknowledgements.
This work is a part of my PhD thesis under the supervisionof Ivo Dell’Ambrogio. 1.
Bicategorical notions
Usual definitions and conventions.
We will follow the naming conventionsof [JY21] for bicategorical notions. In particular, the terms -category and -functor denote the strict ones; bicategory and pseudofunctor refer to the strongbut (possibly) non strict variants. We will use the term (2 , -category for a -category with only invertible -morphisms , and the term (2 , -functor for a -functor between (2 , -categories .We recall some usual constructions and properties of -categories we will use,and introduce some notations.1.1. Notation.
We use the symbol ≃ to denote an isomorphism between two ob-jects (in a -category) and the symbol ∼ = to denote an equivalence between twoobjects (in a -category).1.2. Definition.
Let C be a 2-category. The opposite -category C op of C is the -category with: • Objects: the objects of C • Hom-categories: C op ( A, B ) = C ( B, A ) Notation.
We use the notation [ A , B ] for the -category of pseudofunctors,pseudonatural transformations and modifications, between two -categories A and B .1.4. Notation.
Let I , C be -categories and T be an object of C . The constant2-functor I → C with value T is denoted by ∆ T .1.5. Definition.
Let I , C be -categories and D : I → C be a -functor. A (pseudo)bilimit of D is an object L of C and a family of equivalences Ψ T : C ( T, L ) ∼ = [ I , C ](∆ T, D ) pseudonatural in T . When it exists, the bilimit of D is unique up to equivalenceand the object L is noted bilim I D .We call the canonical pseudonatural transformation Ψ L (Id L ) the standard cone of the bilimit L .1.6. Definition.
Let C be a -category. Let f : a → c and g : b → c be two -morphisms of C with the same target. A -pullback of f and g is a bilimit of thediagram: a c b f g CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 3
Notation.
We write ( f | g ) for the -pullback of f and g . We will use thefollowing naming scheme for the structural - and -morphisms: ca b ( f | g ) f ( f | g ) ∼ g ( f | g ) ( f | g ) Definition.
Let C be a (2 , -category. Fix a (2 , -functor F : I → C and anobject c of C . The slice F /c is the (2 , -category with: • Objects: the pairs ( i, f ) consisting of an object i of I and a morphism f : F i → c • Morphisms ( i, f ) → ( i ′ , f ′ ) : the pairs ( u, µ ) consisting of a morphism u : i → i ′ of I and a 2-isomorphism µ : f → f ′ F ( u ) of C : F i F i ′ c f F u f ′ µ • ( u, µ ) ⇒ ( v, ν ) : the 2-morphisms α : u ⇒ v of I satisfying: F i F i ′ c F v F uf f ′ µ F α = F i F i ′ c F vf f ′ ν • Compositions are induced by the compositions of I and C .A slice -category F /c is endowed with a canonical forgetful -functor: F /c → I ( i, f ) i ( u, µ ) uα α Adjunctions and monads.
We recall the definitions of adjunctions and mon-ads, and some of their basic properties in
Cat .1.9.
Definition.
Let
C, D be two categories. An adjunction between C and D is a quadruple ( ℓ, r, η, ǫ ) of a functor ℓ : C → D , a functor r : D → C , a naturaltransformation η : Id C ⇒ rℓ and a natural transformation ǫ : ℓr ⇒ Id D , such that:(1.10) Id ℓ = ǫℓ ◦ ℓη (1.11) Id r = rǫ ◦ ηr We note an adjunction ℓ ⊣ r , with the natural transformations η and ǫ omitted.The natural transformation η is called the unit of the adjunction and the naturaltransformation ǫ is called the counit . JUN MAILLARD
Definition.
Let C be a category. A monad on C is a monoid object inthe monoidal category Cat ( C, C ) of endofunctors of C , that is, a triple ( T , η, µ ) consisting of a functor T : C → C , a natural transformation η : Id C → T and anatural transformation µ : T ◦ T ⇒ T such that the following diagrams commute: T ◦ T ◦ T T ◦ TT ◦ T T T µµ T µµ T ◦ T T T ◦ TT µ T η η T µ Proposition.
Let
C, D be two categories and ( ℓ : C → D, r : D → C, η, ǫ ) bean adjunction ℓ ⊣ r between C and D . Then the triple ( rℓ, η, rǫℓ ) defines a monadon C . Definition.
Let C be a category and T be a monad on C . The Eilenberg-Moore category C T of T is the category with: • Objects: the pairs ( c, f ) with c an object of C and f : T c → c a morphismof C . • Morphisms ( c, f ) → ( c ′ , f ′ ) : the morphisms g : c → c ′ such that the follow-ing square commutes: T c T c ′ c c ′ T gf f ′ g • Composition is induced by the composition of C .There is a canonical functor U : C T → C ( c, f ) cg g and a natural transformation ¯ µ : T U ⇒ U with components ¯ µ ( c,f ) = f Definition.
Let C be a category and ( T , η, µ ) be a monad on C . A left module on T , or left T -module, is a category D endowed with a functor V : D → C and anatural transformation ν : T V ⇒ V such that the following diagrams commute TT V T V T V V µV T ν νν V T VV ηVν Definition.
Let C be a category and ( T , η, µ ) be a monad on C . The -category of left T -modules is the category with: • Objects: the left T -modules. • -morphisms ( D, V, ν ) → ( E, W, ω ) : the pairs ( X, χ ) consisting of a mor-phism X : D → E and a natural isomorphism χ : V ∼ −→ W X such that: χ ◦ ν = ωX ◦ T χ CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 5 • -morphisms ( X, χ ) ⇒ ( X ′ , χ ′ ) : the -morphisms ψ : X → X ′ such that: D EC X ′ XV Wψ χ = D EC X ′ V Wχ ′ Proposition.
Let C be a category and T be a monad on C . The Eilenberg-Moore category C T endowed with U and ¯ µ (definition 1.14) is a left module on T (definition 1.15).Moreover it is terminal among the left modules on T in the following sense: if ( D, V, ν ) is another left module on T , there is a unique functor K : D → C T suchthat V = U K and ν = ¯ µK . Remark.
The left T -module ( C T , U, ¯ µ ) of proposition 1.17 is pseudo biterminalin the -category of left T -modules (definition 1.16): for any other left T -module ( D, V, ν ) , there is a -morphism ( K, κ ) : (
D, V, ν ) → ( C T , U, ¯ µ ) , unique up to aunique -isomorphism.1.3. String diagrams.
The proof of our main result is an extensive manipulationof string diagrams. We recall the general ideas of string diagrams, and introduceour notations. A fully detailed description of string diagrams can be found in [JY21,§3.7].String diagrams are used to depict natural transformations. They are the dualdiagrams of the “usual” pasting diagrams. • An object A is represented by a labeled surface A • A 1-morphism f : A → B is represented by a labeled vertical edge fA B with the source on the left and the target on the right. • A 2-morphism φ : f ⇒ g : A → B is represented by a labeled vertex fφgA B with the source above and the target below. The identity 1-morphisms maybe omitted. JUN MAILLARD • We will occasionally represent identity 2-morphisms by a white dot, thatis, the string diagram fgA B is the identity between the 1-morphisms f and g . • When dealing with an adjunction ℓ ⊣ r , units and counits will be repre-sented by black dots. For instance, the string diagram ℓ r is the unit of the adjunction ℓ ⊣ r .1.19. Remark.
The notation for the unit and the counit of an adjunction ℓ ⊣ r givesa simple interpretation of the unit-counit laws (eq. (1.10) and eq. (1.11)): ℓℓ = ℓℓ rr = r r -categories as -categories. A -category can be seen as a (2 , -category,with only identities as -morphisms, and a -functor between -categories can itselfbe viewed as a -functor. Hence all the constructions on -categories can be appliedto -categories and -functors. Reciprocally there are several ways to extract -categories from bicategories.1.20. Definition.
Let C be a bicategory with only invertible 2-morphisms. The truncated category τ C is the category where: • Objects are the objects c of C . • Morphisms c → c ′ are the classes of -morphisms c → c ′ of C up to -isomorphism. • Composition is induced by the composition of C .There is a canonical projection pseudofunctor π : C → τ C .1.21. Proposition.
Let C be a bicategory with only invertible -morphisms and D be a -category seen as a -category. Let F : C → D be a pseudofunctor. Thenthere is a unique -functor τ F : τ C → D factoring F through π : C τ C D F π τ F Definition.
Let C be a (2 , -category. The underlying category C (1) is thecategory where: CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 7 • Objects are the objects c of C . • Morphisms c → c ′ are the -morphisms of C . • Composition is induced by the composition of C .There is a canonical inclusion -functor ι : C (1) → C .1.23. Proposition.
Let C be a (2 , -category and D be a -category. Let F : C → D be a -functor. Then the bilimit of F , if it exists, can be expressed as either ofthe following two ordinary limits in D : bilim C F ≃ lim τ C τ F ≃ lim C (1) F ◦ ι A 2-categorical framework for fusion theory
The classical Cartan-Eilenberg formula.
Definition.
Let G be a finite group and S be a p -Sylow subgroup of G . The fusion system F S ( G ) of G is the category with: • Objects: the subgroups P of S . • Morphisms P → Q : the group morphisms P → Q induced by the conjuga-tion action of G : F S ( G )( P, Q ) = { c g : p gpg − | g ∈ G, gP g − ⊂ Q } . • Composition is the composition of group morphisms.The fusion system F S ( G ) of a group G is canonically endowed with a forgetfulfunctor to the category gp of finite groups: U : F S ( G ) → gp P Pf f The Cartan-Eilenberg formula expresses the cohomology (with trivial coeffi-cients) of a group G as a limit over the fusion system F S ( G ) of G in the category F p Alg gr of graded F p -algebras:2.2. Proposition.
For any finite group G and p -Sylow S of G : H ∗ ( G ; F p ) ≃ lim F S ( G ) op H ∗ ( − ; F p ) ◦ U Definition.
Let G be a finite group and S be a p -Sylow of G . The transportercategory T S ( G ) is the category with: • Objects: the subgroups P of S . • Morphisms P → Q : the elements g of G such that gP g − ⊂ Q : T S ( G )( P, Q ) = { g ∈ G | gP g − ⊂ Q } . • Composition is given by the multiplication of G .The transporter category T S ( G ) of a group G is also canonically endowed witha forgetful functor to the category gp of finite groups: U : F S ( G ) → gp P Pg c g : p gpg − JUN MAILLARD
This forgetful functor U : T S ( G ) → gp factors through F S ( G ) . The comparisonfunctor π : T S ( G ) → F S ( G ) is the identity on objects and full. T S ( G ) F S ( G ) gp πU U Note that the functor π is a final functor, in the sense of [Mac71, §IX.3], since itis the identity on objects and full. Hence, there is a canonical isomorphism: lim F S ( G ) op H ∗ ( − ; F p ) ◦ U ≃ lim T S ( G ) op H ∗ ( − ; F p ) ◦ U In particular, the Cartan-Eilenberg formula can equivalently be stated as: H ∗ ( G ; F p ) ≃ lim T S ( G ) op H ∗ ( − ; F p ) ◦ U The extended transporter (2,1)-category of a group.
We present a -categorification ˆ T S ( G ) of the transporter category. This -category will have a rolesimilar to the one the fusion category F S ( G ) and the transporter category T S ( G ) have in the classical Cartan-Eilenberg formula.Mostly for convenience, rather than restricting our attention to the category offinite groups gp, we should consider the (2 , -category of finite groupoids gpd . Itinherits all its structure of -category from the usual structure of -category on Cat ,the category of small categories, functors and natural transformations. The finitegroups are viewed in gpd as the groupoids with exactly one object. We should alsonote that the -morphisms of gpd are relevant from the point of view of the fusionin groups. Indeed, given two parallel morphisms φ, ψ : H → G between groups, a -morphism φ ⇒ ψ is precisely an element g of G such that: ∀ h ∈ H, φ ( h ) = gψ ( h ) g − We note gpd f the 2-full subcategory of gpd with faithful functors as -morphisms.2.4. Definition.
Let G be a group and S be a p -Sylow of G . Denote by i : S → G the inclusion, seen as a -morphism of gpd f . The extended transporter category ˆ T S ( G ) is the (2 , -category with: • Objects: the pairs ( P, j P : P → S ) of a finite groupoid P and a faithfulfunctor j P : P → S . • ( P, j P ) → ( Q, j Q ) : the pairs ( a, α ) of a faithful functor a : P → Q and -morphisms α : ij Q a ⇒ ij P in gpd f . • ( a, α ) → ( b, β ) : the -morphisms φ : α ⇒ β in gpd f , suchthat: GP Q ij P baφβ ij Q = GP Q ij P aα ij Q • Compositions are induced by the obvious pasting diagrams in gpd f . CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 9
The extended transporter category is endowed with a forgetful -functor: U : ˆ T S ( G ) → gpd f ( P, j P ) P ( a, α ) aφ φ Remark.
Since i : S → G is a faithful functor between 1-object groupoids, thefactorization of a functor i P : P → G through i , if it exists, is unique. Hence theobjects of the transporter -category ˆ T S ( G ) could equivalently be described as pairs ( P, i P : P → G ) such that the faithful functor i P factors through i : i P = ij P for some j P : P → S Remark.
The underlying -category ˆ T S ( G ) (1) is quite similar to the classi-cal T S ( G ) . Actually, the main difference is the addition of finite coproducts. Insection 4, we choose not to distinguish them. Since we are working with product-preserving functors (over the opposite categories), this is not really a problem.2.7. Remark.
Biequivalently, the extended transporter category ˆ T S ( G ) can be de-fined as follows, using generic constructions of bicategories. Consider the slice(2,1)-category gpd f /G . Then we can define the extended transporter category ˆ T S ( G ) as the 2-category of subobjects of ( S, i ) in gpd f /G , that is, the full, 2-fullsubcategory of gpd f /G over objects ( P, i P : P → G ) such that i P factors (up tosome isomorphism) through i : S → G : P SG i P ∃ ∼ = i In this construction, the forgetful -functor U : ˆ T S ( G ) → gpd f appears as the restriction of the forgetful -functor gpd f /G → gpd f .2.8. Remark.
Note that there is a canonical way to recover the group G from the -functor U as a pseudo bicolimit: G ∼ = bicolim ˆ T S ( G ) U Group representations as bilimits
Group representations as Mackey 2-functors.
In this section, k is acommutative Z ( p ) -algebra (that is, every prime integer different from p is invertiblein k ), for instance a field of characteristic p .3.1. Notation.
Let G be a finite group. • The category mod ( k G ) is the category of k -linear representations of finitedimension of the group G . • The category stmod ( k G ) is the stable category of k -linear representationsof finite dimension of the group G , for k a field of characteristic p . • The category D b ( k G ) is the (bounded) derived category of the finite di-mensional k G -modules. The three mappings G mod ( k G ) , G stmod ( k G ) and G D b ( k G ) ex-tend to -functors M : ( gpd f ) op → Add from the opposite of the -category offinite groupoids (with faithful -morphisms), to the -category of small additivecategories. The image M ( f ) of a group morphism f : H → G , noted f ∗ , is the restriction along f ; similarly given a -morphism α : f ⇒ g in gpd f , we note α ∗ : f ∗ ⇒ g ∗ the image M ( α ) . We can see that these three -functors are cohomo-logical Mackey -functors with values in k -linear idempotent-complete categories , aswe now explain.3.2. Definition. A Mackey -functor is a -functor M : ( gpd f ) op → Add satisfyingthe following four axioms:(Mack 1)
Additivity: If i G : G → G ⊔ H and i H : H → G ⊔ H are the canonicalinclusions into a coproduct of groupoids, then the induced functor ( i ∗ G , i ∗ H ) : M ( G ⊔ H ) → M ( G ) × M ( H ) is an equivalence.(Mack 2) If f : H → G is a -morphism in gpd f , its image f ∗ has both a left adjoint f ! and a right adjoint f ∗ : f ! ⊣ f ∗ ⊣ f ∗ (Mack 3) Beck-Chevalley property:
For any 2-pullback square in gpd f (see nota-tion 1.7) K G ( f | g ) L f λ ( f | g ) ( f | g ) g the mates λ ! : ( f | g ) ! ( f | g ) ∗ ⇒ g ∗ f ! of λ ∗ and ( λ − ) ∗ : g ∗ f ∗ ⇒ ( f | g ) ∗ ( f | g ) ∗ of ( λ − ) ∗ are invertible, where:(3.3) λ ! = ( f | g ) ∗ ( f | g ) ! f ! g ∗ λ ∗ λ ∗ = f ∗ g ∗ ( f | g ) ∗ ( f | g ) ∗ ( λ − ) ∗ (Mack 4) Ambidexterity:
For any -morphism f : H → G in gpd f , the left and rightadjoints of f ∗ are isomorphic: f ! ≃ f ∗ Notation. A -functor M : ( gpd f ) op → Cat which only satisfies the axioms(Mack 2) and (Mack 3) is said to have the
Beck-Chevalley property .A -functor M : ( gpd f ) op → Cat which only satisfies one half of the axioms(Mack 2) and (Mack 3) is said to have the left Beck-Chevalley property or rightBeck-Chevalley property , accordingly. CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 11
Definition. A cohomological Mackey -functor is a Mackey -functor M suchthat for any inclusion of groups i : H → G the composite natural transformation Id M ( G ) i ∗ i ∗ ≃ i ! i ∗ Id M ( G ) η ǫ is equal to [ G : H ] σ , for some -isomorphism σ . In particular, if M takes valuesin k -linear categories, the composite natural transformation ǫη above is invertiblewhenever [ G : H ] is coprime to p , since k is a Z ( p ) -algebra.The following comparison theorem between M ( G ) and the associated Eilenberg-Moore category will appear in [BD]; we reproduce here its short proof.3.6. Theorem.
Let M be a cohomological Mackey -functor (definition 3.2 anddefinition 3.5) with values in k -linear idempotent-complete categories.Then, for any finite group G and subgroup i : H → G with [ G : H ] coprime to p , the adjunction i ! ⊣ i ∗ is monadic. That is, the canonical comparison functor M ( G ) → M ( H ) T is an equivalence, where T = i ∗ i ! is the monad induced by i .Proof. Recall that ǫ : i ! i ∗ ⇒ Id M ( G ) is the counit of the adjunction i ! ⊣ i ∗ and η : Id M ( G ) ⇒ i ∗ i ∗ is the unit of the adjunction i ∗ ⊣ i ∗ .Since M is cohomological and [ G : H ] is coprime to p , the natural transformation ǫη is invertible. Hence ǫ admits a section, and the adjunction i ! ⊣ i ∗ is monadic, by[Bal15, Lemma 2.10]. (cid:3) This naturally leads to the following definition:3.7.
Definition. A p -monadic Mackey -functor is a Mackey -functor M suchthat, for any inclusion of groups i : H → G with index coprime to p , the canonicalcomparison functor M ( G ) → M ( H ) T is an equivalence, where T = i ∗ i ! is the monad induced by i .3.8. Remark.
By theorem 3.6, any cohomological Mackey -functor with values in k -linear idempotent-complete categories is p -monadic. This includes the -functorsmod ( k − ) , stmod ( k − ) and D b ( k − ) ; other examples can be found in [BD20, Chap-ter 4].For instance, the mapping G coMack k ( G ) associating to a group G the cate-gory of cohomological G -local Mackey 1-functor extends to a cohomological Mackey -functors. This is a corollary of [BD20, Proposition 7.3.2] applied to the cohomo-logical Mackey -functor S ( G ) = perm k ( G ) of permutations k G -modules and theYoshida theorem[Web00], characterizing the category of cohomological Mackey 1-functors as coMack k ( G ) ∼ = Fun + ( perm k ( G ) , Ab ) . The main theorem.
Theorem.
Let M : ( gpd f ) op → Add be a p-monadic Mackey 2-functor (defi-nition 3.7). Then for any group G and any p -Sylow S of G , there is an equivalence M ( G ) ∼ = bilim ˆ T S ( G ) op M ◦ U where ˆ T S ( G ) is the extended transporter -category of G (definition 2.4). Using the p -monadicity, theorem 3.9 is a corollary of the following result:3.10. Theorem.
Let M : ( gpd f ) op → Cat be a -functor with the left Beck-Chevalleyproperty (notation 3.4).For any finite group G with a finite p -Sylow i : S → G , there is an equivalence M ( S ) T ∼ = bilim ˆ T S ( G ) op M ◦ U where T = i ∗ i ! is the monad induced by i . Remark.
The equivalence of theorem 3.10 is reminiscent of the Bénabou-Roubaud theorem [BR70]. This is actually an instance of a generalization to -categories; a precise statement will appear in my thesis [Mai]. Nonetheless, theessential arguments of the proof are already present in this article.We will now construct the equivalence of theorem 3.10. Let L = bilim ˆ T S ( G ) op M ◦ U and note L • its standard cone (definition 1.5). In the rest of this section, we will: • construct a functor A : L → M ( S ) T , • construct a functor B : M ( S ) T → L , and • show that the functors A and B are mutual pseudoinverses.This will prove theorem 3.10.3.3. Construction of a comparison functor L → M ( S ) T . Recall that M ( S ) T has a canonical structure of left module on T consisting of the forgetful functor U : M ( S ) T → M ( S ) and a natural transformation ¯ µ : T U ⇒ U . Moreover, it isterminal among the left modules on T (proposition 1.17).One can define a structure of left T -module for the bilimit L as follows. Thereis an obvious functor L → M ( S ) , namely the component L S of the standard cone.The definition of the action ν use the invertibility of the mate λ ! (see (Mack 3)):(3.12) L S T L S ν = L S i ! i ∗ λ − L λ L S ( i | i ) ! ( i | i ) ∗ CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 13 where λ = ( i | i ) is defined by the following 2-pullback square in gpd f GS S ( i | i ) i λ i ( i | i ) ( i | i ) and L λ is the composite -morphism ( i | i ) ∗ L S L ( i | i ) L ( i | i ) ( i | i ) ∗ L S L (( i | i ) , Id) L (Id ,λ ) L − i | i ) , Id)
Proposition.
The action ν is unital.Proof. First, note that there is a diagonal morphism ∆ : S → ( i | i ) satisfying ( i | i )∆ = ( i | i )∆ = Id S and λ ∆ = Id i . We thus have the following relation: i ! i ∗ = i ! i ∗ λ ∗ ∆ ∗ = i ! i ∗ λ ∗ ∆ ∗ = i ! i ∗ λ ! ∆ ∗ (3.14) Hence, we have: L S L S ν (3.12)and (3.14) = L S L S λ ! λ − L λ ∆ ∗ = L S L S L λ ∆ ∗ = L S L S (cid:3) Proposition.
The action ν is associative.Proof. We introduce another 2-pullback square S ( i | i ) ( i | i ) X = (( i | i ) | ( i | i )) ( i | i ) κ ( i | i ) v w The universal property of ( i | i ) allows us to define a functor ∇ : X → ( i | i ) such that(3.16) v ( i | i ) iw ( i | i ) iλκ λi = v ( i | i ) iw ( i | i ) iλ ∇ CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 15
Applying the 2-functor M on eq. (3.16) yields(3.17) v ∗ ( i | i ) ∗ i ∗ w ∗ ( i | i ) ∗ i ∗ λ ∗ κ ∗ λ ∗ i ∗ = v ∗ ( i | i ) ∗ i ∗ w ∗ ( i | i ) ∗ i ∗ λ ∗ ∇ ∗ Similarly, the compatibility between the cone L and the 2-morphisms of ˆ T S ( G ) provides the equation(3.18) v ∗ ( i | i ) ∗ L S w ∗ ( i | i ) ∗ L S L λ κ ∗ L λ L S = v ∗ ( i | i ) ∗ L S w ∗ ( i | i ) ∗ L S L λ ∇ ∗ We can now prove the associativity of ν . We want to prove :(3.19) L S i ! i ∗ i ! i ∗ L S ν ν = L S i ! i ∗ i ! i ∗ L S ν Consider the mate κ ! of κ ∗ : κ ! = v ∗ w ! ( i | i ) ! ( i | i ) ∗ κ ∗ Since λ ! and κ ! are invertible, eq. (3.19) is equivalent to the equation:(3.20) L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ! λ ! λ ! ν ν = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ! λ ! λ ! ν Expanding the definitions of ν and κ ! on the left-hand side of eq. (3.20) leads tothe following computation: L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ! λ ! λ ! ν ν (3.12) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ! λ ! λ ! λ − λ − L λ L λ (3.3) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ∗ L λ L λ CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 17 = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S L λ κ ∗ L λ (3.18) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S L λ ∇ ∗ To simplify the right-hand side of eq. (3.20), first note that we have the followingrelation: L S ( i | i ) ∗ L S ( i | i ) ∗ λ ∗ ν = L S ( i | i ) ∗ L S ( i | i ) ∗ λ ! ν (3.12) = L S ( i | i ) ∗ L S ( i | i ) ∗ L λ = L S ( i | i ) ∗ L S ( i | i ) ∗ L λ (3.21)Then, by expanding the definitions of λ ! and κ ! in the right-hand side of eq. (3.20),we get: L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ! λ ! λ ! ν (3.3) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S κ ∗ λ ∗ λ ∗ ν = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S λ ∗ κ ∗ λ ∗ ν (3.17) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S λ ∗ ν ∇ ∗ (3.21) = L S ( i | i ) ∗ v ∗ w ! ( i | i ) ! L S L λ ∇ ∗ Thus eq. (3.20) holds, and so does eq. (3.19). (cid:3)
By the universal property of M ( S ) T (proposition 1.17) applied to the left T -module ( L, L S , ν ) , we have:3.22. Proposition.
There is a unique functor A : L → M ( S ) T fitting in the diagram L M ( S ) T M ( S ) A L S U and satisfying ¯ µA = ν . The comparison functor M ( S ) T → L . In the reverse direction, giving afunctor B : M ( S ) T → L is equivalent (definition 1.5) to giving a pseudonaturaltransformation φ : ∆ M ( S ) T ⇒ M ◦ U between 2-functors ˆ T S ( G ) → Cat . Definethe components of φ as follows: • for any object ( P, j P ) of ˆ T S ( G ) , φ ( P,j P ) : M ( S ) T M ( S ) M ( P ) U j ∗ P CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 19 • for any -morphism ( a, α ) : ( P, j P ) → ( Q, j Q ) , that is a -morphism a : P → Q and a -morphism α : ij Q a ⇒ ij P in gpd f ,(3.23) φ ( a,α ) = U j ∗ Q a ∗ α ∗ ¯ µU j ∗ P Proposition.
The natural transformation φ ( a,α ) is invertible.Proof. Consider ψ ( a,α ) = U j ∗ P ( α ∗ ) − ¯ µU j ∗ Q a ∗ We can check, using the associativity and the unitality of the action ¯ µ , and thedefinition of the multiplication µ of T , that ψ ( a,α ) ◦ φ ( a,α ) = Id and φ ( a,α ) ◦ ψ ( a,α ) =Id . (cid:3) Proposition.
The family of functors and natural transformations φ is apseudonatural transformation φ : ∆ M ( S ) T ⇒ M ◦ U .Proof. This is done by the following straightforward computations. φ (Id , Id) = U j ∗ P Id ∗ ¯ µ j ∗ P = U j ∗ P ¯ µU j ∗ P = U j ∗ P U j ∗ P = Id For any composable morphisms ( a, α ) : ( P, j P ) → ( Q, j Q ) and ( b, β ) : ( Q, j Q ) → ( R, j R ) ,using the associativity of the action ¯ µ and the definition of the multiplication µ of the monad T : φ ( a,α ) ◦ a ∗ φ ( b,β ) = U j ∗ R b ∗ a ∗ β ∗ ¯ µ α ∗ ¯ µU j ∗ P = U j ∗ R b ∗ a ∗ β ∗ µ α ∗ ¯ µU j ∗ P = U j ∗ R b ∗ a ∗ β ∗ α ∗ ¯ µU j ∗ P = U j ∗ R b ∗ a ∗ α ∗ ◦ a ∗ β ∗ ¯ µU j ∗ P = φ ( b,β ) ◦ ( a,α ) For any -morphism ζ : ( a, α ) ⇒ ( b, β ) between parallel morphisms ( P, j P ) → ( Q, j Q ) in ˆ T S ( G ) , we compute: φ ( b,β ) ◦ φ ( Q,j Q ) ( ζ ∗ ) = U j ∗ Q a ∗ ζ ∗ β ∗ ¯ µU j ∗ P = U j ∗ Q a ∗ α ∗ ¯ µU j ∗ P = φ ( a,α ) These equalities show the pseudonaturality of φ . (cid:3) Hence, by the universal property of the bilimit:3.26.
Proposition.
There exists a unique (up-to 2-isomorphism) functor B : M ( S ) T → L and a modification m : φ ∼ = L · B . CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 21
The functors A and B are mutually pseudoinverse. Proposition.
The composite AB : M ( S ) T → M ( S ) T is isomorphic to theidentity functor.Proof. The universal property (remark 1.18) of M ( S ) T guarantees that (Id M ( S ) T , Id U ) is the unique -endomorphism of the left T -module ( M ( S ) T , U, ¯ µ ) , up to a -isomorphism. Hence, the existence of a -isomorphism ξ : U ∼ −→ U AB such that ( AB, ξ ) : ( M ( S ) T , U, ¯ µ ) → ( M ( S ) T , U, ¯ µ ) is a -morphism of left T -modules (defi-nition 1.16) implies that Id M ( S ) T and AB are isomorphic. Let us find such a ξ .First, as φ S = j ∗ S U = U we can take the -isomorphism m S : U AB = L S B ∼ = φ S = U given by proposition 3.26 as our -isomorphism ξ . Moreover, since m : L B → φ isa modification, m S ◦ ¯ µAB (3.22) = m S ◦ νB (3.12) = B L S i ∗ i ∗ λ − L λ m S U (3.26) = B L S i ∗ i ∗ λ − m S φ (Id ,λ ) U (3.23) = B L S i ∗ i ∗ λ − m S λ ∗ ¯ µU (3.3) = B L S i ∗ i ∗ m S ¯ µU = ¯ µ ◦ T m S This equality shows that ( AB, m S ) : ( M ( S ) T , U, ¯ µ ) → ( M ( S ) T , U, ¯ µ ) is a -morphismof left T -modules. Since such a -morphism is unique up to -isomorphism, AB isisomorphic to Id M ( S ) T . (cid:3) Proposition.
The composite BA : L → L is isomorphic to the identityfunctor.Proof. Similarly we will show that the induced cone ˜ L := L · ( BA ) is isomorphicto the standard cone L , that is, that there is an invertible modification n betweenthem. On any object ( P, j P ) define n P as n P : ˜ L P = L P BA m P A −−−→ φ P A = j ∗ P U A = j ∗ P L S = L P The family of morphisms ( n P ) is an invertible modification between the cones ˜ L and φA . Thus, it suffices to check that φA and L are indeed the same cones.Their components at any object ( P, j P ) coincide; we still have to check that thecomponents at any morphism ( a, α ) are equal.At the morphism (Id , λ ) : (( i | i ) , ( i | i )) → (( i | i ) , ( i | i )) of ˆ T S ( G ) , we have: φ (Id ,λ ) A = A U ( i | i ) ∗ λ ∗ ¯ µA U ( i | i ) ∗ (3.22) = L S ( i | i ) ∗ λ ∗ ν L S ( i | i ) ∗ (3.12) = L S ( i | i ) ∗ λ ∗ λ − L λ L S ( i | i ) ∗ (3.3) = L S ( i | i ) ∗ λ ∗ λ − L λ L S ( i | i ) ∗ (3.3) = L S ( i | i ) ∗ L λ L S ( i | i ) ∗ = L λ For any morphism ( a, α ) : ( P, j P ) → ( Q, j Q ) there is a (unique) morphism ofgroupoids ∇ α : P → ( i | i ) such that ( i | i ) ∇ α = j Q a ( i | i ) ∇ α = j P λ ∇ α = α CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 23
Hence, φ ( a,α ) A = A U j ∗ Q a ∗ α ∗ ¯ µA U j ∗ P = A U j ∗ Q a ∗ λ ∗ ¯ µA U j ∗ P ∇ ∗ α = a ∗ j ∗ Q L S j ∗ P L S L λ ∇ ∗ α = L ( a,α ) Thus, BA is isomorphic to the identity functor. (cid:3) Putting together proposition 3.27 and proposition 3.28, we get:3.29.
Theorem.
The categories L and M ( S ) T are equivalent. Applications
In this section, we describe two different ways of extracting the classical Cartan-Eilenberg formula from theorem 3.9.4.1.
Extracting Hom-sets.
In this subsection, we will express the Hom-sets of abilimit in
Cat for a strict
Cat .4.1.
Definition.
Let I be a (small) -category and D : I → Cat be a -functor.Let L D be the category with: • Objects:
The pairs of families d := (( d i ) i , ( d f ) f ) such that: – for each object i ∈ I , d i is an object of D i – for each -morphism f : i → j in I , d f is an isomorphism D f ( d i ) ∼ −→ d j in D j . – for any -morphism φ : f ⇒ g in I , with f, g : i → j , d g ◦ ( D φ ) d i = d f – for any composable -morphisms f : i → j and g : j → k in I , d g ◦ f = d g ◦ ( D g )( d f ) • Morphisms (( d i ) , ( d f )) → (( d ′ i ) , ( d ′ f )) : the families of morphisms ( δ i : d i → d ′ i ) i ∈ Ob I such that, for any morphism f : i → j in I , the following square commutes:(4.2) D f ( d i ) d j D f ( d ′ i ) d ′ jd f D f ( δ i ) δ j d ′ f • Composition is induced componentwise by the compositions of the cate-gories ( D i ) i .The category L D is endowed with a canonical cone φ over D with components: φ i : L D ⇒ D i (( d i ) i , ( d f ) f ) d i ( f i ) i f i for any object i ∈ I ( φ f ) (( d i ) i , ( d f ) f ) = d f for any morphism f : i → j ∈ I Proposition.
The category L D , with its canonical cone, is a bilimit of D : L D ∼ = bilim I D Proof.
The category L D is an explicit description of the category [∆1 , D ] of the(pseudo)cones over D with vertex , the category with exactly one object and itsidentity morphism.In turn, [∆1 , D ] is a model of the bilimit of D by the following classical equiva-lence, pseudonatural in X ∈ Cat : Cat ( X, [∆1 , D ]) ∼ = Cat (1 , [∆ X, D ]) ∼ = [∆ X, D ] (cid:3) We now fix a -diagram D : I → Cat and two objects d , d ′ of L D . By definition:(4.4) L D ( d, d ′ ) = { ( δ i : d i → d ′ i ) i | the squares (4.2) commute } We can define an associated -diagram on the underlying -category I (1) of I ,obtained by forgetting the -morphisms, as follows:4.5. Definition.
The diagram D d,d ′ : I (1) → Set is: D d,d ′ : (cid:26) i
7→ D i ( d i , d ′ i ) f : i → j
7→ D j ( d − f , d ′ f ) ◦ D f Proposition.
There is an isomorphism: L D ( d, d ′ ) ≃ lim I (1) D d,d ′ Proof.
The limit lim I (1) D d,d ′ has an explicit description, as a limit in Set : lim I (1) D d,d ′ = { ( δ i ∈ D d,d ′ ( i )) i | ∀ f : i → j, D d,d ′ ( f )( δ i ) = δ j } Unfolding the definitions, we precisely get back the set of (4.4). (cid:3)
CATEGORIFICATION OF THE CARTAN-EILENBERG FORMULA 25
Remark.
The expression of the diagram D d,d ′ simplifies when the objects d andd’ are such that, for all f : i → j , d f = d ′ f = Id : D d,d ′ : (cid:26) i
7→ D i ( d i , d ′ i ) f : i → j
7→ D f Example.
We can use these results to recover the Cartan-Eilenberg formulafor the Tate cohomology ˆ H ∗ ( G ; k ) of a finite group G over a field k of characteristicp. Indeed, for any finite group G , the Tate cohomology groups can be recovered asHom-sets of stmod ( k G ) : stmod ( k G )(Ω n k , k ) = ˆ H n ( G ; k ) Fix a finite group G and a p -Sylow S of G . We consider the -diagram: D = stmod ( k − ) ◦ U : ˆ T S ( G ) op → Cat
Note that, for any n , Ω n k ∈ stmod ( k G ) can be seen as an object of L D satisfyingthe conditions of remark 4.7. Hence we have: ˆ H n ( G ; k ) = stmod ( k G )(Ω n k , k ) ≃ lim P ∈T S ( G ) op stmod ( k P )(Ω n k , k )= lim P ∈T S ( G ) op ˆ H n ( P ; k ) The Cartan-Eilenberg formula for the usual group cohomology H ∗ ( G ; k ) can besimilarly recovered from the -functor D b ( k − ) .4.2. Categorical invariants.
Another way to exploit theorem 3.9 is to try tofactor M as a composite of two -functors M : ( gpd f ) op ˜ M −−→ C W −→ Cat where ˜ M is product preserving, W : C → Cat reflects bilimits, and to find abilimit-preserving -functor H : C → D to a -category D . In this case, for any finite group G with p -Sylow S : H ◦ ˜ M ( G ) ≃ H ( bilim ˆ T S ( G ) op ˜ M ◦ U ) ≃ bilim ˆ T S ( G ) op H ◦ ˜ M ◦ U ≃ lim T S ( G ) op H ◦ ˜ M ◦ U (by proposition 1.23)(4.9)4.10. Example.
We sketch an example of application, taking M = stmod ( k − ) . Wecan factor it through the -category Add gr • of graded additive categories with a dis-tinguished object, choosing the trivial module k in each stmod ( k G ) and graduatingby the Heller shift Ω . The forgetful functor Add gr • → Cat reflects bilimits. Thenconsider the well-defined bilimit-preserving 2-functor to the -category of rings: H : Add gr • → Ring ( C, • C ) C ∗ ( • C , • C )( F : C → D, u : F ( • C ) ∼ −→ • D ) D ( u − , u ) ◦ Fα Id Note that, for any finite group G : H ◦ ˜ M ( G ) = stmod ( k G ) ∗ ( k , k ) = ˆ H ∗ ( G ; k ) Hence by eq. (4.9), we obtain the same Cartan-Eilenberg formula as in example 4.8: ˆ H ∗ ( G ; k ) = H ◦ ˜ M ( G ) ≃ lim T S ( G ) H ◦ ˜ M ◦ U = lim P ∈T S ( G ) ˆ H ∗ ( P ; k ) It would be interesting to find other factorizations and bilimit-preserving -functors H with value in a -category for the various p -monadic Mackey 2-functors:this would give possibly new “Cartan-Eilenberg formulas”. References [Bal15] Paul Balmer. “Stacks of Group Representations”. In:
Journal of the Eu-ropean Mathematical Society issn : 1435-9855. doi : .[BD] Paul Balmer and Ivo Dell’Ambrogio. Cohomological Mackey 2-Functors .In preparation.[BD20] Paul Balmer and Ivo Dell’Ambrogio.
Mackey 2-Functors and Mackey 2-Motives . Zuerich, Switzerland: European Mathematical Society Publish-ing House, July 31, 2020. isbn : 978-3-03719-209-2. doi : .[BR70] Jean Bénabou and Jacques Roubaud. “Monades et Descente”. In: C. R.Acad. Sci. Paris Sér. A-B
270 (1970), A96–A98. issn : 0151-0509.[CE56] Henri Cartan and Samuel Eilenberg.
Homological Algebra . Princeton Uni-versity Press, Princeton, N. J., 1956, pp. xv+390.[JY21] Niles Johnson and Donald Yau. . New York:Oxford University Press, 2021. isbn : 978-0-19-887137-8.[Mac71] Mac Lane Saunders.
Categories for the Working Mathematician . Gradu-ate Texts in Mathematics. New York: Springer-Verlag, 1971. isbn : 978-0-387-90036-0.[Mai] Jun Maillard. Ph.D. Thesis (In preparation). Université de Lille.[Mai21] Jun Maillard.
On 2-Final 2-Functors . 2021. arXiv: .[Par17] Sejong Park. “Mislin’s Theorem for Fusion Systems through MackeyFunctors”. In:
Communications in Algebra issn : 0092-7872, 1532-4125. doi : .[Web00] Peter Webb. “A Guide to Mackey Functors”. In: Handbook of Algebra,Vol. 2 . Vol. 2. Handb. Algebr. Elsevier/North-Holland, Amsterdam, 2000,pp. 805–836.
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
Email address : [email protected] URL ::